# 1 The Time Value of Money Copyright by Diane Scott Docking 2014.

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1 The Time Value of Money Copyright by Diane Scott Docking 2014

Learning Objectives 1.To understand the concept of “the time value of money” 2.Be able to mathematically compute: a)Future and Present Values b)Annuities c)An amortization table 3.To understand difference between “compounding” and “discount ing 4.To understand and compute “Effective Annual Return” Copyright by Diane Scott Docking 2014 2

3 Time is Money \$100 in your hand today is worth more or less than \$100 in one year? Money earns interest The higher the interest rate, the faster your money grows Q: How much would \$1,000 promised in one year be worth today if you could be earning 12% interest? Copyright by Diane Scott Docking 2014

4 Time is Money A: \$1,000/(1.12) 1 = \$892.86 Copyright by Diane Scott Docking 2014 t0t0 t1t1 \$1,000 PV Verify: \$892.86 x.12 = \$ 107.14 interest earned plus initial investment = \$ 892.86 yields = \$1,000.00

5 4 Major TVM Problems Deal with four different types of problems Single Amounts (Lump sum) Future value Present value Annuities (multiple amounts) Future value—advance or arrears Present value—advance or arrears Copyright by Diane Scott Docking 2014

6 4 Major TVM Problems Mathematics For each type of problem an equation will be presented Time lines Graphic portrayal of a time value problem 012 Copyright by Diane Scott Docking 2014

7 1. FV of a Lump Sum Future Value An amount to be accumulated in the future by investing currently or over time at a certain investment % rate The compounded value of a sum is its future value Compounding is earning interest on interest For example: Invest \$100 today for 3 years @ 10%. Future value is _____? \$100 (1.10) 3 = \$133.10 Copyright by Diane Scott Docking 2014

8 Example: What’s the FV of an initial \$100 investment after 3 years if i = 10%? FV = ? 0123 10% 100 Finding FV is compounding. Copyright by Diane Scott Docking 2014

9 Graph: What’s the FV of an initial \$100 investment after 3 years if i = 10%? FV = ? 0123 10% 100 Copyright by Diane Scott Docking 2014 \$100 x.10 = Interest \$ 10 Principal \$100 Balance \$110 x.10 = interest \$11 balance \$110 Balance \$121 x.10 = interest \$ 12.10 balance \$121.00 FV = \$133.10 \$100 x.10 = \$10 x 3 yrs. = \$30 in interest if withdraw interest each year.

10 Formula—Future Value The future value (FV) of an amount How much a sum of money placed at interest (i) will grow into in some period of time If the time period is one year –FV 1 = PV + iPV or FV 1 = PV(1+i) If the time period is two years –FV 2 = FV 1 + iFV 1 or FV 2 = PV(1+i) 2 If the time period is generalized to n years –FV n = PV(1+i) n Copyright by Diane Scott Docking 2014

11 Another Example: FV of a Lump Sum Mary invests \$1,000 today for 20 years @ 8%. How much will she have in 20 years? FV n = PV(1+i) n FV 20 = \$1,000 (1.08) 20 = \$4,660.96 Copyright by Diane Scott Docking 2014

12 2. PV of a Lump Sum Present Value The amount invested today to have a future \$ sum at a certain interest % rate The discounted value of a sum is its present value For example: Want to have \$100 in 3 years. Current interest rate is 10%. How much must you invest today? Present value is _____? \$100 /(1.10) 3 = \$75.13 Copyright by Diane Scott Docking 2014

13 Example: How much would you have to invest today (i.e. PV) if i = 10% to end up with \$100 (i.e. FV) after 3 years? FV = 100 0123 10% PV = ? Finding PV is discounting. Copyright by Diane Scott Docking 2014

14 10% Or another way to say it: What’s the PV of \$100 due in 3 years if i = 10%? Finding PVs is _____, and it’s the reverse of __________. What puts money into the future? 100 0123 PV = ? Copyright by Diane Scott Docking 2014 easy future value The present?

15 Formula for Present Value of an Amount The future and present values factors are reciprocals of one another. Copyright by Diane Scott Docking 2014 FV n = PV(1+i) n Recall our formula for FV: Using algebra:

16 Graph: What’s the PV if i = 10% to end up with FV = \$100 after 3 years? FV = 100 0123 10% 75.13 Copyright by Diane Scott Docking 2014 Verify: \$75.13 x.10 = Interest \$ 7.51 Principal \$75.13 Balance \$82.64 x.10 = interest \$ 8.26 balance \$82.64 Balance \$90.90 x.10 = interest \$ 9.09 balance \$90.90 FV = \$99.99 = \$100 \$100 /(1.10) 3 = \$75.13

17 Annuity due Payments in advance Payments at beginning of interest periods Regular annuity Ordinary annuity Payments in arrears Payments at end of interest periods Annuity Vocabulary Lab All mean the same thing Copyright by Diane Scott Docking 2014

18 Ordinary Annuity PMT 0123 i% PMT 0123 i% PMT Annuity Due What’s the difference between an ordinary annuity and an annuity due? Copyright by Diane Scott Docking 2014

19 3. FV of an annuity Future Value An amount to be accumulated in the future by investing at regular intervals over time at a certain investment % rate Copyright by Diane Scott Docking 2014

20 Example 1: What is the FV of a 3-year ordinary annuity of \$100 at 10%? \$100100 0123 10% FV= 10% Copyright by Diane Scott Docking 2014 110 121 100(1.10)= 100(1.10) 2 = \$331

21 Example 2: What’s the FV of a 3-year annuity due of \$100 at 10%? 100 \$100 100 0123 10% FV = Copyright by Diane Scott Docking 2014 100(1.10)= 110 100(1.10) 2 = 121 133.10 100(1.10) 3 = \$364.10

22 Relationship between FVs of ordinary annuity and annuity due. 10% FV= 10% Copyright by Diane Scott Docking 2014 100 0123 110 121 100(1.10)= 100(1.10) 2 = 331 Ordinary annuity x (1.10) Annuity Due FV = 364.10

23 4. PV of an annuity Present Value The lump sum amount to be invested today at a certain investment % rate to yield a desired payment at regular intervals into the future. Copyright by Diane Scott Docking 2014

24 Example 1: What’s the PV of an 3-year ordinary annuity of \$100 per year at 10%? 100 0123 10% = PV Copyright by Diane Scott Docking 2014 =100/(1.10) 1 90.91 =100/(1.10) 2 82.64 75.13 \$248.68 =100/(1.10) 3

25 Example 2: What’s the PV of an 3-year annuity due of \$100 per year at 10%? 100 0123 10% 100 = PV Copyright by Diane Scott Docking 2014 90.91 =100/(1.10) 82.64 =100/(1.10) 2 273.55

26 Relationship between PVs of ordinary annuity and annuity due. = PV Copyright by Diane Scott Docking 2014 \$248.68 100 0123 10% =100/(1.10) 90.91 =100/(1.10) 2 82.64 75.13 =100/(1.10) 3 x(1.10) \$273.55 = PV Ordinary annuity Annuity due

27 Payments with uneven cash flows Investments can have irregular cash flows Future Value An amount to be accumulated in the future by investing varying amounts at regular intervals over time at a certain investment % rate Present Value The lump sum amount to be invested today at a certain investment % rate to yield desired payments of varying amounts at regular intervals into the future. Copyright by Diane Scott Docking 2014

28 Example: What is the FV of this uneven cash flow stream if rates are 10% and payment is made at end of the year? 0 100 1 300 2 3 10% -50 4 Copyright by Diane Scott Docking 2014 300 x (1.10) = 300 x (1.10) 2 = 100 x (1.10) 3 = 330 363 133.10 FV = \$776.10

29 Example: What is the PV of this uneven cash flow stream if rates are 10% and payment is made at the end of the year? 0 100 1 300 2 3 10% -50 4 90.91 ______ \$530.08 = PV Copyright by Diane Scott Docking 2014 =100/(1.10) =300/(1.10) 2 =300/(1.10) 3 =-50/(1.10) 4 247.93 225.39 -34.15

30 Agree to make “step payments” on a contract as follows: \$1,000 per month for the first year \$1,300 per month for the second year \$2,000 per month for the third year At 10% annual rate, what is the PV of these payments? Example: PV of Monthly Uneven Cash Flow Stream Copyright by Diane Scott Docking 2014

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32 Will the FV of a lump sum be larger or smaller if we compound more often, holding the I/Y% constant? Why? _______? If compounding is more frequent than once a year–for example, daily,—then, interest is earned on interest more often That means more frequent compounding will produce a higher future value. Frequency of Compounding Copyright by Diane Scott Docking 2014 Larger

33 0123 10% 0123 5% 456 100 123 0 Annually: FV 3 = 100(1.10) 3 = Semiannually: FV 6 = 100(1.05) 6 = Frequency of Compounding Copyright by Diane Scott Docking 2014 \$133.10 \$134.01

34 When we look at “compounding” over periods less than 1 year, we are looking at “effective” rates. We deal with 3 different rates: i Nom = nominal rate i Per = periodic rate EAR= EFF = effective annual rate Effective Annual Rates (EARs) Copyright by Diane Scott Docking 2014

35 i Nom is stated in contracts. Aka: nominal, or stated, or quoted rate per year. Periods per year (m) must also be given. Examples: n 8%; Quarterly n 8%; Daily interest (365 days) n Nominal rate equals periodic rate times # of periods in a year 1. Nominal Interest Rates Copyright by Diane Scott Docking 2014

36 Periodic rate = i Per = i Nom /m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 365 for daily compounding. 2. Periodic Rates Copyright by Diane Scott Docking 2014

37 Example: 8% quarterly: i Per = 8%/4 = 2%. 8% daily (365): i Per = 8%/365 = 0.021918%. Example: 10% semi-annually: i Per = 10%/2 = 5%. 10% monthly: i Per = 10%/12 = 0.8333%. Examples of: Periodic Rates Copyright by Diane Scott Docking 2014

38 3. Effective Annual Rates Copyright by Diane Scott Docking 2014 Where: m is number of compounding periods per year.

39 Example: EAR% for a nominal rate of 8%, compounded quarterly? Daily? Copyright by Diane Scott Docking 2014

40 Example: EAR% for a nominal rate of 10%, compounded semiannually? Copyright by Diane Scott Docking 2014

41 Effective Annual Rate: EAR = EFF% of 10% EAR Annual = 10.0% EAR S = = % EAR Q == % EAR M == % EAR D == % 10.25 10.381 10.471 10.516 Copyright by Diane Scott Docking 2014 (1.05) 2 - 1 (1.025) 4 - 1 (1.0833333) 12 - 1 (1.000273973) 365 - 1 (1.10) 1 - 1

Finding “n” and “i” of a Lump Sum Copyright by Diane Scott Docking 2014 42

Finding “n”, “i”, and “payment” of Annuities It is more difficult to find “ n ”, “ i ”, and “payments” FV & PV of an ordinary annuity:  Example 1: At what interest rate would you have to invest \$100 payments made annually (at end of year) to end up with \$331 in 3 years?  Example 2: If you invest \$100 annually (at end of year) at 10%, for how many years must you invest before your future value equals \$331?  Example 3: How much must you invest annually (at end of year) at 10%, for 3 years to accumulate a future value of \$331 at the end of 3 years? Copyright by Diane Scott Docking 2014 43

44 RECALL: What is the FV of a 3-year ordinary annuity of \$100 at 10%? \$100100 0123 10% FV= 10% Copyright by Diane Scott Docking 2014 110 121 100(1.10)= 100(1.10) 2 = \$331

Finding “n”, “i”, and “payment” of Annuities Finding “ i ” FV & PV of an ordinary annuity:  Example 1: At what interest rate would you have to invest \$100 payments made annually (at end of year) to end up with \$331 in 3 years? \$100 + \$100(1+ i) 1 + \$100(1+ i) 2 = \$331 \$100(1+ i) 1 + \$100(1+ i) 2 = \$231 (1+ i) 1 + (1+ i) 2 = 2.31 1+ i + 1+ 2i + i 2 = 2.31 3i + i 2 = 0.31 i(3+ i) = 0.31Then through trial and error solve for i. At i = 10%, the equation is solved. Copyright by Diane Scott Docking 2014 45

Finding “n”, “i”, and “payment” of Annuities Finding “ n ” FV & PV of an ordinary annuity:  Example 2: If you invest \$100 annually (at end of year) at 10%, for how many years must you invest before your future value equals \$331?  Trial and error: \$100 + \$100(1.10) 1 = \$210 so 2 years is NOT long enough \$100 + \$100(1.10) 1 + \$100(1.10) 2 = 210 + 121 = \$331 At 3 years, the equation is solved. Copyright by Diane Scott Docking 2014 46

Finding “n”, “i”, and “payment” of Annuities Finding “ payment ” FV & PV of an ordinary annuity:  Example 3: How much must you invest annually (at end of year) at 10%, for 3 years to accumulate a future value of \$331 at the end of 3 years? X + X(1.10) 1 + X(1.10) 2 = \$331 X [1+ 1.10 + 1.21] = \$331 X [3.31] = \$331 X = \$331 / 3.31 X = \$100 Copyright by Diane Scott Docking 2014 47

Finding “n”, “i”, and “payment” of Annuities It is more difficult to find “ n ”, “ i ”, and “ payments” FV & PV of an annuity due:  Example 1: At what interest rate would you have to invest \$100 payments made annually (at beginning of year) to end up with \$364.10 in 3 years?  Example 2: If you invest \$100 annually (at beginning of year) at 10%, for how many years must you invest before your future value equals \$364.10?  Example 3: How much must you invest annually (at beginning of year) at 10%, for 3 years to accumulate a future value of \$364.10 at the end of 3 years? Copyright by Diane Scott Docking 2014 48

49 RECALL: What’s the FV of a 3-year annuity due of \$100 at 10%? 100 0123 10% FV = Copyright by Diane Scott Docking 2014 100(1.10)= 110 100(1.10) 2 = 121 133.10 100(1.10) 3 = 364.10

Finding “n” and “i” Finding “ i ” FV & PV of an annuity due:  Example 1: At what interest rate would you have to invest \$100 payments made annually (at beginning of year) to end up with \$364.10 in 3 years? \$100(1+ i) 1 + \$100(1+ i) 2 + \$100(1+ i) 3 = \$364.10 \$100[(1+ i) 1 + (1+ i) 2 + (1+ i) 3 ]= \$364.10 (1+ i) 1 + (1+ i) 2 + (1+ i) 3 = 3.641 1+ i +1+ 2i + i 2 + 1+ 3i + 3i 2 + i 3 = 3.641 3+6i + 4i 2 + i 3 = 3.641 i(6 + 4i + i 2 ) = 0.641 Then through trial and error solve for i. At i = 10%, the equation is solved. Copyright by Diane Scott Docking 2014 50

Finding “n” and “i” Finding “ n ” FV & PV of an annuity due:  Example 2: If you invest \$100 annually (at beginning of year) at 10%, for how many years must you invest before your future value equals \$364.10?  Trial and error: \$100(1.10) 1 = \$110 so 1 years is NOT long enough \$100(1.10) 1 + \$100(1.10) 2 = \$110 + \$121 = \$231 so 2 years is NOT long enough \$100(1.10) 1 + \$100(1.10) 2 + \$100(1.10) 3 = 110 + 121 + 133.10 = \$364.10 At 3 years, the equation is solved. Copyright by Diane Scott Docking 2014 51

Finding “n”, “i”, and “payment” of Annuities Finding “ payment ” FV & PV of an annuity due:  Example 3: How much must you invest annually (at beginning of year) at 10%, for 3 years to accumulate a future value of \$364.10 at the end of 3 years? X(1.10) 1 + X(1.10) 2 + X(1.10) 3 = \$364.10 X [1.10 + 1.21 + 1.331] = \$364.10 X [3.641] = \$364.10 X = \$364.10 / 3.641 X = \$100 Copyright by Diane Scott Docking 2014 52

53 Amortization What it is: equal, periodic repayment on a loan that reflects part interest and part principal over the life of the loan Copyright by Diane Scott Docking 2014

54 Amortization Schedules Construct an amortization schedule for a \$1,000, 10% annual rate loan with 3 equal payments in arrears (paid at end of year). Copyright by Diane Scott Docking 2014

55 Step 1: Find the required payments. PMT 0123 10% -1,000 Amortization Schedule Copyright by Diane Scott Docking 2014

56 Step 2: Find interest charge for Year 1. INT t = Beg bal t (i) INT 1 = \$1,000(0.10) = \$100. Step 3: Find repayment of principal in Year 1. PRN = PMT - INT = \$402.11 - \$100 = \$302.11. Amortization Schedule Copyright by Diane Scott Docking 2014

57 Step 4: Find ending balance after Year 1. End balance = Beg balance - PRN = \$1,000 - \$302.11 = \$697.89. Repeat these steps for Years 2 and 3 to complete the amortization table (see next page). Amortization Schedule Copyright by Diane Scott Docking 2014

58 BEGPRNEND YRBALPMTINTPMTBAL 1\$1,000\$402.11\$100\$302.11\$697.89 2697.89 402.1169.79332.32365.57 3365.57 402.1336.56365.570 TOT1,206.35206.351,000 Amortization Schedule—cont’d. Copyright by Diane Scott Docking 2014 May have to adjust last payment for rounding differences and to get ending balance to zero.

59 \$ 0123 402.11 Interest 302.11 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is Principal Payments Time Copyright by Diane Scott Docking 2014

60 Amortization tables are widely used in loan business—housing, autos, etc. Financial calculators (and spreadsheets) are great for setting up amortization tables. Are a mechanism for “proving” the return or providing early payoff information. Amortization—cont’d. Copyright by Diane Scott Docking 2014